Forecasting the weather or the climate is not an easy matter. It requires modelling of numerous natural phenomena and interaction between several sciences, ranging from mathematics to biology, via computer science, physics and chemistry.

Whether it concerns manoeuvring a plane, or the mechanical resistance of a complicated structure, or managing automobile traffic, progress in all these areas doesn't come from purely technological inventions alone. It also involves abstract research, like the mathematical theory of control..

A ruler, a pencil, cardboard, scissors and glue: one doesn't need more to give a mathematician pleasure, and present interesting problems whose study often turns out to be useful in other areas, in totally unexpected ways.

How to escape being detected by radar? What is the optimal shape of a sound-proof wall? Can one improve the quality of sonographic images? To get a satisfactory answer, these questions require a thorough theoretical analysis.

Scientists are not the only ones to be inspired by mathematics. Many artists have drawn the subject of some of their works from it. The converse is also sometimes true, as in the case of perspective, where art has led to geometrical theories.

The biological activity of a DNA molecule depends mainly on the way it is arranged in space and the way in which it is twisted - things which fall within the province of the mathematical theory of knots.

Throughout their history, mathematics and philosophy have had a close and enigmatic relationship. It would be necessary to go as far back as Plato in ancient Greece and Descartes at the dawn of modern times. Let us cite here two great figures of the 20th century, David Hilbert and Edmund Husserl.

Problems of organisation and planning faced by an airline company are similar to those met with in other sectors of industry. Operations research, with which tens of thousands of mathematicians and engineers in the world concern themselves, tries to solve these problems as well as possible.

Physicists have aspired for a long time to building a theory which would cover all the elementary particles and all their interactions. Since about twenty years ago, they have a trail that is promising. But to explore it, they must navigate in highly abstract spaces where even mathematicians had not yet ventured.

The financial world fixes the price of options by means of formulas which have been obtained thanks to relatively recent workin mathematics. The search for better formulas continues... and small-time speculators are not the only ones who are interested!

For detecting and correcting the inevitable errors which creep in during digital information transmission, specialists in coding theory make use of abstract methods which arise from algebra or geometry.

Reconstructing a surface from the knowledge of only some of its points: a problem that one comes across often, be it in geological exploration, in recording archaeological remains, or in medical or industrial imaging.

At the end of the 19th century, there were very few ``geometers'', as mathematicians were formerly called. In one century, their numbers have augmented considerably. Today, they are facing profound changes in their discipline.